Learning collective variables that respect permutational symmetry.

Area of Science:

• Statistical mechanics
• Computational physics
• Materials science

Background:

• Coarse-grained models are crucial for understanding systems with identical interacting particles.
• These models aid in identifying metastable states, describing dynamics, and estimating transition rates.
• Permutational symmetry, alongside translational and rotational symmetries, is a key characteristic of such systems.

Purpose of the Study:

• To develop a numerical framework for learning collective variables that inherently respect translational, rotational, and permutational symmetries.
• To accurately estimate transition rates and residence times in systems with identical interacting particles.
• To provide a robust method for coarse-grained modeling of complex particle systems.

Main Methods:

• A novel framework combining sort-based featurization and residence manifold learning.
• Utilizing autoencoders with an orthogonality relationship-based loss function to learn symmetric collective variables.
• Employing the committor of the reduced model as a reaction coordinate for forward flux sampling and transition path sampling control.

Main Results:

• The proposed framework successfully learns collective variables that preserve system symmetries.
• Computed transition rates and residence times using the reduced models show excellent agreement with brute-force methods.
• Demonstrated efficacy through case studies on Lennard-Jones-7 (2D) and Lennard-Jones-8 (3D) systems.

Conclusions:

• The developed numerical framework provides an accurate and efficient method for coarse-grained modeling of systems with permutational symmetry.
• This approach enhances the understanding of metastable states, dynamics, and transition pathways in particle clusters.
• The method offers a significant advancement in computational techniques for complex interacting particle systems.